nLab integer Heisenberg group

Idea

The integer Heisenberg groups or discrete Heisenberg groups are non-abelian discrete central extensions of the integer product groups $\mathbb{Z}^{2g}$, $g \in \mathbb{N}$, analogous to how the ordinary Heisenberg groups are non-abelian central extensions of $\mathbb{R}^{2g}$. Generally there are analogous $R$-Heisenberg groups for any commutative ring $R$.

In fact, for even multiples of the basic such extension, integer Heisenberg groups are discrete subgroups of ordinary $\mathbb{R}$-Heisenberg groups covering a lattice inclusion $\mathbb{Z}^{2g} \hookrightarrow \mathbb{R}^{2g}$ (cf. Prop. below).

In physics, specifically in quantum field theory, these even-weight integer Heisenberg groups are closely related to quantum abelian Chern-Simons theory (spelled out below, following SS25, see also at 2-Cohomotopy moduli of surfaces).

Definition

Integer Heisenberg group extensions of $\mathbb{Z}^2$

We discuss the restriction of the basic $\mathbb{R}$-Heisenberg group $H_3$ (see there) along the inclusion $\mathbb{Z}^2 \hookrightarrow \mathbb{R}^2$ and the mod-$n$ quotient groups of that.

In the following, we denote cyclic groups, for $n \in \mathbb{N}$, uniformly by

$$\mathbb{Z}_{n} \;\coloneqq\; \left\{ \begin{array}{lcl} \mathbb{Z} &\vert& n = 0 \\ \mathbb{Z}/(n) &\vert& \text{otherwise} \mathrlap{\,.} \end{array} \right.$$

Recall (see there) the equivalence of central extensions of a discrete group $G$ by an abelian group $A$ (for the present case equipped with the trivial $G$-action) with the degree=2 group cohomology $H^2(G;A)$.

Proposition

For $n \in \mathbb{N}$, the central group extensions of $\mathbb{Z}^2$ by $\mathbb{Z}_n$ are all integral multiples of the following elementary Heisenberg extension:

(1)$$\left\{ \begin{array}{l} \big(a,b,[n]\big) \,\in\, \mathbb{Z} \times \mathbb{Z} \times \mathbb{Z}_{2\vert k \vert} \,, \\ \big( a,b,[n] \big) \cdot \big( a',b',[n'] \big) \;=\; \big( a + a', b+ b', [ n + n' + {\color{red} a \, b' } ] \big) \end{array} \right\} \,.$$

Proof

The central extensions in question

$$1 \to \mathbb{Z}_n \xrightarrow{cntrl} H \longrightarrow \mathbb{Z}^2 \to 1$$

are classified by the degree=2 group cohomology of $\mathbb{Z}^2$ with coefficients in $\mathbb{Z}_n$ (equipped with the trivial $\mathbb{Z}^2$-action).

That second group cohomology is:

(2)$$\begin{array}{ccl} H^2_{Grp}(\mathbb{Z}^2;\, \mathbb{Z}_n) &\simeq& H^2(B \mathbb{Z}^2;\, \mathbb{Z}_n) \\ &\simeq& H^2(T^2;\, \mathbb{Z}_n) \\ &\simeq& H^2(S^1_a \vee S^1_b \vee S^2;\, \mathbb{Z}_n) \\ &\simeq& H^2(S^2;\, \mathbb{Z}_n) \\ &\simeq& \mathbb{Z}_n \mathrlap{\,,} \end{array}$$

where we used, in order of the appearance:

1. that group cohomology is ordinary cohomology of the domain group’s classifying space;

2. that the classifying space of the integers is the circle and that this respects Cartesian products;

3. that the stable homotopy type of the 2-torus is the wedge sum of two circles with a 2-sphere (see there).

The generator of this cohomology group must be the group 2-cocycle

$$\alpha \,\in\, H^2_{Grp}\big( \mathbb{Z}^2 ;\, \mathbb{Z}_n \big)$$

given by

(3)$$\begin{array}{ccc} \mathbb{Z}^2 \times \mathbb{Z}^2 &\xrightarrow{\; \alpha \;}& \mathbb{Z}_{n} \\ \big( (a,b), (a',b') \big) &\mapsto& [a b'] \mathrlap{\,,} \end{array}$$

whose cocycle condition $[a b'] + [(a+a')b''] = [a(b'+b'')] + [a' b'']$ is evidently satisfied due to the distributive law in the ring $\mathbb{Z}$ and which is clearly not divisible.

Example

In the integer Heisenberg group (1)

• the inverse elements are given by

  $$\big(a ,\, b ,\, [n]\big)^{-1} \;=\; \big( -a ,\, -b ,\, [-n + a b] \big)$$

• the basic group commutators give

  (4)$$\begin{array}{ccl} \big[ (1,0,0), (0,1,0) \big] &\equiv& (1,0,0) \cdot (0,1,0) \cdot (1,0,0)^{-1} \cdot (0,1,0)^{-1} \\ &=& (1,1,1) \cdot (-1,0,0) \cdot (0,-1,0) \\ &=& (0,1,1) \cdot (0,-1,0) \\ &=& (0,0,1) \mathrlap{\,.} \end{array}$$

Of particular interest is the second multiple of this basic extension:

Proposition

Twice the basic integer Heisenberg extension (3) at $n = 2k$ is isomorphic to

(5)$$\left\{ \begin{array}{l} \big(a, b, [n]\big) \,\in\, \mathbb{Z} \times \mathbb{Z} \times \mathbb{Z}_{2\vert k\vert} \,, \\ \big( a, b, [n] \big) \cdot \big( a', b', [n'] \big) \;=\; \big( a + a' ,\, b + b' ,\, [n + n' + {\color{red} a \cdot b' - a' \cdot b}] \big) \end{array} \right\} \mathrlap{\,.}$$

Proof

It is sufficient to see that the second red summand in (5) is a group 2-cocycle cohomologous to the first red summand, which means equivalently that the 2-cocycle

$$\big( (a, b), (a', b') \big) \;\mapsto\; a \cdot b' + a' \cdot b$$

has a coboundary: This coboundary is readily found to be

$$(a, b) \;\mapsto\; - a \cdot b \,,$$

since

$$- a \cdot b - a' \cdot b' \,+\, (a + a') \cdot (b + b') \;=\; a \cdot b' + a' \cdot b \,.$$

Properties

Basic properties

Proposition

The multiple=2 discrete Heisenberg group $\mathbb{Z}_{2{\vert k \vert}} \to \widehat{\mathbb{Z}^{2}} \to \mathbb{Z}^2$ (5) is the following quotient group of the product group $F(a,b) \times \mathbb{Z}_{2{\vert k \vert}}$ with the free group $F(a,b)$ on two generators:

(6)$$\widehat{\mathbb{Z}^2} \,\simeq\, \big( F(a,b) \times \mathbb{Z}_{2 {\vert k \vert}} \big) \big/ \big( a \cdot b = [2] \cdot b \cdot a \big) \,.$$

Proof

The map

$$\begin{array}{ccc} \frac{ F(a,b) \times \mathbb{Z}_{2 {\vert k \vert}} }{ a \cdot b = [2] \cdot b \cdot a } &\xrightarrow{\;}& \widehat{\mathbb{Z}^2} \\ [a] &\mapsto& (1,0,0) \\ [b] &\mapsto& (0,1,0) \\ [n] &\mapsto& (0,0,[n]) \end{array}$$

is clearly a bijection on underlying sets, and is a group homomorphism since the quotient relation (6) is respected in $\widehat{\mathbb{Z}^2}$, by (4).

(cf. also arXiv:2203.08030, p 21)

Linear representations

Example

For $k \in \mathbb{Z}$, the non-trivial irrep of twice the $\mathbb{Z}_{2 {\left\vert k \right\vert}}$-Heisenberg extension of $\mathbb{Z}^2$ (5)

(7)$$\begin{array}{ccc} \widehat{\mathbb{Z}^2} & \xrightarrow{\; W \;} & \mathrm{Aut}\big( \mathscr{H}_1 \big) \end{array}$$

is, up to isomorphism, given by:

$$\mathscr{H}_1 \;\coloneqq\; \mathbb{C}\big[\mathbb{Z}_{2{\vert k \vert}}\big]$$

$$\begin{array}{ccccccl} W_a & \coloneqq & W(1,0,0) &\colon& {\big\vert [n] \big\rangle} &\mapsto& e^{2 \pi \mathrm{i} \tfrac{n}{k}} {\big\vert [n] \big\rangle} \\ W_b &\coloneqq& W(0,1,0) &\colon& {\big\vert [n] \big\rangle} &\mapsto& {\big\vert [n + 1] \big\rangle} \\ \zeta &\coloneqq& W(0,0,1) &\colon& {\big\vert [n] \big\rangle} &\mapsto& e^{\pi \mathrm{i} \tfrac{1}{k}} {\big\vert [n] \big\rangle} \mathrlap{\,.} \end{array}$$

For instance:

(8)$$W_a \cdot W_b \cdot \zeta^{-1} \;=\; W(1,1,0) \;=\; W_b \cdot W_a \cdot \zeta^{+1} \,.$$

Proof

To see that this is a linear representation, by (6) is it sufficient to check that the basic group commutator is represented, in that

(9)$$W_a \cdot W_b \,=\, e^{ 2 \pi \mathrm{i} \tfrac{1}{k} } \, W_b \cdot \mathrlap{\,,}$$

which is evidently the case, since

$$\begin{array}{ccl} W_a \cdot W_b {\big\vert [n] \big\rangle} &=& W_a \cdot {\big\vert [n+1] \big\rangle} \\ &=& e^{2 \pi \mathrm{i} \tfrac{n{\color{red}+1}}{k}} \, {\big\vert [n+1] \big\rangle} \mathrlap{\,,} \end{array}$$

while

$$\begin{array}{ccl} W_b \cdot W_a {\big\vert [n] \big\rangle} &=& e^{2 \pi \mathrm{i} \tfrac{n}{k}} \, W_b {\big\vert [n] \big\rangle} \\ &=& e^{2 \pi \mathrm{i} \tfrac{n}{k}} \, {\big\vert [n + 1] \big\rangle} \mathrlap{\,.} \end{array}$$

To see that this is irreducible just note that a linear basis of the underlying vector space is obtained by successively acting with $W_b$ on, say ${\big\vert [1] \big\rangle}$.

For more see Floratos & Tsohantjis 2022.

Modular automorphisms

Remark

(relation to quantum states and quantum observables of abelian Chern-Simons theory)
The complex dimension of the irrep $\mathscr{H}_1$ (7) is that expected for the space of quantum states of abelian Chern-Simons theory on a 2-torus (cf. Manoliu 1998a p 40)

$$dim(\mathscr{H}_1) \;=\; k \,,$$

and the group commutator-relation (9)

$$W_a \cdot W_b = e^{2\pi \mathrm{i}\tfrac{1}{k}} \, W_b \cdot W_a$$

of linear operators acting on this irrep (7) reflects the characteristic commutator-relation of Wilson loop-quantum observables in abelian Chern-Simons theory on a 2-torus (cf. Tong 2016 (5.28), p 166).

Moreover, the integer Heisenberg group knows about the modular group acting on this space of quantum states, hence about the “modular functor” of abelian Chern-Simons theory:

Definition

(modular action on the integer Heisenberg group)
Since the colored summand in (5) is the canonical symplectic form on $\mathbb{Z}^2$, the integer symplectic group in dimension 2, hence the modular group

$$\Big\{ g \in \mathrm{GL}_2(\mathbb{Z}) \,\Big\vert\, \underset{ \mathrm{det}(g) \cdot (a b' - a' b) }{ \underbrace{ (g_{1 1} a + g_{12} b) (g_{2 1} a' + g_{22} b') - (g_{11} a' + g_{12}b') (g_{21} a + g_{22} b) } } = a b' - a'b \Big\} \;\simeq\; \mathrm{SL}_2(\mathbb{Z}) \,,$$

acts by evident group automorphism on (5):

(10)$$\begin{array}{ccc} \mathrm{SL}_2(\mathbb{Z}) \times \widehat{\mathbb{Z}^2} &\xrightarrow{}& \widehat{\mathbb{Z}^2} \\ \Big( g, \big(a,b,[n]\big) \Big) &\mapsto& \Big( g_{11}a + g_{12}b ,\, g_{21}a + g_{22}b ,\, [n] \Big) \mathrlap{\,.} \end{array}$$

Remark

Recall that the modular group $SL_2(\mathbb{Z}) \subset GL_2(\mathbb{Z})$ is the subgroup of the general linear group generated by the two elements

(11)$$S \;\coloneqq\; \left[ \begin{array}{rr} 0 & 1 \\ -1 & 0 \end{array} \right] \;\;\;\;\; \text{and} \;\;\;\;\; T \;\coloneqq\; \left[ \begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array} \right] \mathrlap{\,.}$$

Proposition

There exists a linear representation

$$\begin{array}{ccc} \mathrm{SL}_2(\mathbb{Z}) \times \mathscr{H}_1 &\xrightarrow{\;}& \mathscr{H}_1 \mathrlap{\,,} \end{array}$$

of the modular group on the underlying complex vector space of (7), which intertwines the action (7) of $\widehat{\mathbb{Z}^2}$ on $\mathscr{H}_1$, with its automorphic images under the modular gorup action (10) on the Heisenberg group, in that:

$$m(W) \cdot m\Big( {\big\vert [n] \big\rangle} \Big) \;=\; m\Big( W \cdot {\big\vert [n] \big\rangle} \Big) \,, \;\;\;\;\; \forall \left\{ \begin{array}{ccc} m &\in& SL_2(\mathbb{Z}) \\ W &\in& \widehat{\mathbb{Z}^2} \\ {\left\vert [n] \right\rangle} &\in& \mathscr{H}_1 \mathrlap{\,.} \end{array} \right.$$

This representation is just the modular action known from abelian Chern-Simons theory at level $k$ (cf. Manoliu 1998a p 67):

$$\begin{array}{ccr} S\Big( {\big\vert [n] \big\rangle} \Big) &=& \frac{1}{\sqrt{\vert k \vert}} \sum_{ [\widehat n] } \, e^{ 2 \pi \mathrm{i} \tfrac{ \widehat{n} \, n }{ k } } {\big\vert [\widehat{n}] \big\rangle} \\ T\Big( {\big\vert [n] \big\rangle} \Big) &=& e^{ \pi \mathrm{i} \tfrac{ n^2 }{ k } } {\big\vert [ n ] \big\rangle} \end{array}$$

Proof

By unwinding the definitions and direct computation we find that the statement holds for $m = S,T$ any one of the modular generators (11):

$$\begin{array}{ccl} S(W_a) \cdot S\Big( {\big\vert [n] \big\rangle} \Big) &\equiv& W_b^{-1} \frac{1}{\sqrt{\vert k \vert}} \sum_{ [\widehat n] } \, e^{ 2 \pi \mathrm{i} \tfrac{ \widehat{n} \, n }{ k } } {\big\vert [\widehat{n}] \big\rangle} \\ &=& \frac{1}{\sqrt{\vert k \vert}} \sum_{ [\widehat n] } \, e^{ 2 \pi \mathrm{i} \tfrac{ (\widehat{n} + 1) \, n }{ k } } {\big\vert [\widehat{n}] \big\rangle} \\ &=& e^{ 2 \pi \mathrm{i} \tfrac{n}{k} } \, S\Big( {\big\vert [n] \big\rangle} \Big) \\ &=& S\Big( W_a {\big\vert [n] \big\rangle} \Big) \mathrlap{\,,} \end{array}$$

$$\begin{array}{ccl} S(W_b) \cdot S\Big( {\big\vert [n] \big\rangle} \Big) &\equiv& W_a \frac{1}{\sqrt{\vert k \vert}} \sum_{ [\widehat n] } \, e^{ 2 \pi \mathrm{i} \tfrac{ \widehat{n} \, n }{ k } } {\big\vert [\widehat{n}] \big\rangle} \\ &=& \frac{1}{\sqrt{\vert k \vert}} \sum_{ [\widehat n] } \, e^{2 \pi \mathrm{i} \tfrac{n}{k}} e^{ 2 \pi \mathrm{i} \tfrac{ \widehat{n} \, n }{ k } } {\big\vert [\widehat{n}] \big\rangle} \\ &=& S\Big( W_a {\big| [n] \big\rangle} \Big) \mathrlap{\,,} \end{array}$$

$$\begin{array}{ccl} T(W_a) \cdot T\Big( {\big\vert [n] \big\rangle} \Big) &\equiv& W_a \, e^{ \mathrm{i} \pi \tfrac{n^2}{k} } {\big\vert [n] \big\rangle} \\ &=& e^{2 \pi \mathrm{i} \tfrac{n}{k}} e^{ \mathrm{i} \pi \tfrac{n^2}{k} } {\big\vert [n] \big\rangle} \\ &=& T\Big( W_a \, {\big\vert [n] \big\rangle} \Big) \mathrlap{\,,} \end{array}$$

and finally, using (8):

$$\begin{array}{ccl} T(W_b)\cdot T\Big( {\big\vert [n] \big\rangle} \Big) &\equiv& W_b \, W_a \, e^{2\pi \mathrm{i} \tfrac{1}{k}} \, e^{\pi \mathrm{i} \tfrac{n^2}{k}} {\big\vert [n] \big\rangle} \\ &=& e^{\pi \mathrm{i} \tfrac{ n^2 + 2n + 1 }{k}} {\big\vert [n + 1] \big\rangle} \\ &=& e^{\pi \mathrm{i} \tfrac{ (n+1)^2 }{k}} {\big\vert [n + 1] \big\rangle} \\ &=& T\Big( W_b {\big\vert [n] \big\rangle} \Big) \mathrlap{\,.} \end{array}$$

But since $S , T \in SL_2(\mathbb{Z})$ generate the whole group (Rem. ), this completes the proof.

Related entries

• Heisenberg group

• 2-Cohomotopy moduli of surfaces

Literature

Generally on the integer/discrete Heisenberg $\mathbb{Z}$-extensions of $\mathbb{Z}^{2g}$:

• Soo Teck Lee, Judith A. Packer: The Cohomology of the Integer Heisenberg Groups, Journal of Algebra 184 1 (1996) 230-250 [doi:10.1006/jabr.1996.0258]

  (concerning its group cohomology)

• Roman Budylin: Conjugacy classes in discrete Heisenberg groups, Sbornik: Mathematics 205 8 (2014) 1069–1079 [arXiv:1405.5499, doi:10.1070/SM2014v205n08ABEH004410]

• Daniel Bump, Persi Diaconis, Angela Hicks, Laurent Miclo, Harold Widom: An Exercise (?) in Fourier Analysis on the Heisenberg Group, Annales de la Faculté des sciences de Toulouse: Mathématiques, Serie 6, Volume 26 (2017) no. 2, pp. 263-288 [arXiv:1502.04160, numdam:AFST_2017_6_26_2_263_0]

• Jayadev S. Athreya, Ioannis Konstantoulas: Lattice deformations in the Heisenberg group, Groups, Geometry and Dynamics 14 3 (2020) 1007–1022 [arXiv:1510.01433, doi:10.4171/ggd/572]

• Uri Bader, Vladimir Finkelshtein, §5 of: On the horofunction boundary of discrete Heisenberg group, Geom Dedicata 208 (2020) 113–127 [arXiv:1904.11234, doi:10.1007/s10711-020-00513-x]

• Ruiwen Dong, p 9 of: Recent advances in algorithmic problems for semigroups, ACM SIGLOG News 10 4 (2023) 3-23 [arXiv:2309.10943, doi:10.1145/3636362.3636365]

On (invertibility in) the group algebra:

• Martin Göll, Klaus Schmidt, Evgeny Verbitskiy: A Wiener Lemma for the discrete Heisenberg group: Invertibility criteria and applications to algebraic dynamics, Monatsh Math 180 (2016) 485–525 [doi:10.1007/s00605-016-0894-0, arXiv:1603.08225]

On their representation theory with an eye towards quantum information theory:

• E. Floratos, I. Tsohantjis: Complete set of unitary irreps of Discrete Heisenberg Group $H W_{2^s}$ [arXiv:2210.04263]

On the automorphism group:

• D.V. Osipov: Discrete Heisenberg group and its automorphism group, Mathematical Notes 98 1 (2015) 185-188 [arXiv:1505.00348, doi:10.1134/S0001434615070160]

In relation to $U(1)$-current algebra (WZW-model):

• Ludwig D. Fadeev: Discrete Heisenberg-Weyl Group and modular group, Lett Math Phys 34 (1995) 249–254 [doi:10.1007/BF01872779]

The above discussion of irreps and modular automorphisms related to abelian Chern-Simons theory follows:

• Hisham Sati, Urs Schreiber: Engineering of Anyons on M5-Probes via Flux-Quantization, Lecture Notes (2025)